nLab space of finite subsets




topology (point-set topology, point-free topology)

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topological homotopy theory



For XX a topological space and nn \in \mathbb{N} a natural number, the space of finite subsets of cardinality n\leq n in XX is the suitably topologized set of finite subsets SXS \subset X of cardinality |S|n\left\vert S\right\vert \leq n, often denoted exp nX\exp^n X or similar (e.g. Félix-Tanré 10).


Relation to unordered configuration space of points

The topological subspace of finite subsets of cardinality exactly equal to nn is the unordered configuration space of points in XX

(1)Conf n(X)exp n(X) Conf_n(X) \hookrightarrow \exp^n(X)

Let XX be an non-empty regular topological space and n2n \geq 2 \in \mathbb{N}.

Then the injection (1)

(2)Conf n(X)exp n(X)/exp n1(X) Conf_n(X) \hookrightarrow \exp^n(X)/\exp^{n-1}(X)

of the unordered configuration space of n points of XX into the quotient space of the space of finite subsets of cardinality n\leq n by its subspace of subsets of cardinality n1\leq n-1 is an open subspace-inclusion.

Moreover, if XX is compact, then so is exp n(X)/exp n1(X)\exp^n(X)/\exp^{n-1}(X) and the inclusion (2) exhibits the one-point compactification (Conf n(X)) +\big( Conf_n(X) \big)^{+} of the configuration space:

(Conf n(X)) +exp n(X)/exp n1(X). \big( Conf_n(X) \big)^{+} \;\simeq\; \exp^n(X)/\exp^{n-1}(X) \,.

(Handel 00, Prop. 2.23, see also Félix-Tanré 10)


  • David Handel, Some Homotopy Properties of Spaces of Finite Subsets of Topological Spaces, Houston Journal of Mathematics, Electronic Edition Vol. 26, No. 4, 2000 (pdfhjm:Vol26-4)

  • Yves Félix, Daniel Tanré Rational homotopy of symmetric products and Spaces of finite subsets, Contemp. Math 519 (2010): 77-92 (pdf)

  • Jacob Mostovoy, Rustam Sadykov, On the connectivity of finite subset spaces (arXiv:1203.5180)

Last revised on October 17, 2019 at 14:45:27. See the history of this page for a list of all contributions to it.